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Spherical Barycentric Coordinates [article]

Torsten Langer, Alexander Belyaev, Hans-Peter Seidel
2006 Symposium on geometry processing : [proceedings]. Symposium on Geometry Processing
We develop spherical barycentric coordinates.  ...  Furthermore, spherical barycentric coordinates can be used for all applications where only planar barycentric coordinates were available up to now.  ...  It follows that there is a bijection between planar barycentric coordinates and spherical barycentric coordinates.  ...

Whitney Forms for Spherical Triangles I: The Euler, Cagnoli, and Tuynman Area Formulas, Barycentric Coordinates, and Construction with the Exterior Calculus [article]

David W. Fillmore, Jay P. Fillmore
2014 arXiv   pre-print
We introduce the three barycentric coordinates of a point within the spherical triangle as area ratios which sum to unity.  ...  The barycentric coordinates are the Whitney 0-forms, scalar functions over the domain of the triangle associated with each vertex.  ...  The Whitney Forms of a Spherical Triangle Let λ A (X) = S A (X) S , λ B (X) = S B (X) S and λ C (X) = S C (X) S be the barycentric coordinates of the point X interior to or on the boundary of ABC, such  ...

Fast Normal Vector Compression with Bounded Error [article]

E. J. Griffith, M. Koutek, Frits H. Post
2007 Symposium on geometry processing : [proceedings]. Symposium on Geometry Processing
The second uses fixed-precision barycentric coordinates. For both, we provide fast (de)compression algorithms and a rigorous upper bound on compression error.  ...  We discuss the effects of base polyhedra on the error bound and suggest polyhedra derived from spherical coverings.  ...  The next two integers represent the u and v barycentric coordinates. Since the barycentric coordinates u, v and w sum to 1, there is no need to explicitly store w.  ...

Generating N-point spherical configurations with low mesh ratios using spherical area coordinates [article]

Brian Hamilton
2021 arXiv   pre-print
The method extends Caspar-Klug icosahedral point-grids to non-icosahedral nets through the use of planar barycentric coordinates, which are subsequently interpreted as spherical area coordinates for spherical  ...  This short contribution presents a method for generating N-point spherical configurations with low mesh ratios.  ...  Homogenizing with spherical area coordinates In order to improve the uniformity of the generated point sets, we can reinterpret the planar barycentric coordinates as a form of spherical barycentric coordinates  ...

Deuteron States Model

Gudrun Kalmbach HE
2018 Journal of Lasers Optics & Photonics
Barycentrical coordinates are generated to determine a nucleons barycentre [2] . 3.  ...  It is postulated that the weak and the spherical, barycentrical coordinates of the strong nucleon interaction are in special relativistic motion towards one another. 5.  ...  Barycentrical coordinates are generated to determine a nucleons barycentre [2] . 3.  ...

Mean value coordinates in 3D

Michael S. Floater, Géza Kós, Martin Reimers
2005 Computer Aided Geometric Design
Trivariate barycentric coordinates can be used both to express a point inside a tetrahedron as a convex combination of the four vertices and to linearly interpolate data given at the vertices.  ...  These coordinates generalize in a natural way a recently constructed set of coordinates for planar polygons, called mean value coordinates.  ...  Clearly, for any p ∈T , we can express the vector e = p − v as e = τ i (e)e i + τ j (e)e j + τ k (e)e k , where τ i , τ j , τ k are the spherical barycentric coordinates of e.  ...

Temporally-smooth Antialiasing and Lens Distortion with Rasterization Map [article]

Jakub Maximilian Fober
2020 arXiv   pre-print
To correct barycentric vector, rst distance pass ⃗ is obtained using inverse of the dot-product between spherical-barycentric coordinate ⃗ and the inversed vertex-distance vector.  ...  Spherical-barycentric vector ⃗ requires perspective-correction to be used as vertex-data interpolation weight.  ...  coordinates . xy , p o l y g [ 1 ] . xy ) , p o l y g [ 2 ] . xy ) −0.5 , 40 max ( max ( p o l y g [ 0 ] . xy , p o l y g [ 1 ] . xy ) , p o l y g [ 2 ] . xy ) +0.5 ) ; 41 } 42 / / Rasterize triangle  ...

Interpolation on spherical geodesic grids: A comparative study

Maria Francesca Carfora
2007 Journal of Computational and Applied Mathematics
Most operational models in atmospheric physics, meteorology and climatology nowadays adopt spherical geodesic grids and require "ad hoc" developed interpolation procedures.  ...  (a) In [1] spherical barycentric coordinates (b 1 , b 2 , b 3 ) of a point P are introduced as the trihedral coordinates of its position vector v with respect to the trihedron generated by the position  ...  (d) In [15] the weights for the bilinear interpolation are the barycentric coordinates evaluated in a local longitude-latitude coordinate system ( , ): if a point P ( , ) belongs to the spherical triangle  ...

Inverse spherical surfaces

G. Casciola, S. Morigi
2005 Journal of Computational and Applied Mathematics
These surfaces can be seen as single-valued functions in spherical coordinates.  ...  In this paper we describe a new way to design rational parametric surfaces defined on spherical triangles which are useful for modelling in a spherical environment.  ...  We now can reinterpret Algorithm 2 as a de Casteljau algorithm which uses different (spherical) barycentric coordinates for each sub-triangle T ( ) ij k instead of using the same barycentric coordinates  ...

A Generic and Scalable Pipeline for GPU Tetrahedral Grid Rendering

Joachim Georgii, Rudiger Westermann
2006 IEEE Transactions on Visualization and Computer Graphics
At the core of our method is the idea to perform the sampling of tetrahedral elements along the view rays entirely in local barycentric coordinates.  ...  This ray is sampled on a spherical slice with distance zs from the eye point. The barycentric coordinate of the sample point is obtained by setting t as the depth of the actual spherical slice, zs.  ...  First, for every element the matrix required to transform eye coordinates into local barycentric coordinates is computed.  ...

Fundamentals of spherical parameterization for 3D meshes

Craig Gotsman, Xianfeng Gu, Alla Sheffer
2003 ACM Transactions on Graphics
We describe a generalization of the method of barycentric coordinates for planar parameterization which solves the spherical parameterization problem, prove its correctness by establishing a connection  ...  Parameterizing a triangle mesh onto the sphere means assigning a 3D position on the unit sphere to each of the mesh vertices, such that the spherical triangles induced by the mesh connectivity are not  ...  Generalizing the barycentric coordinates theory to spherical embeddings is not straightforward.  ...

Fundamentals of spherical parameterization for 3D meshes

Craig Gotsman, Xianfeng Gu, Alla Sheffer
2003 ACM SIGGRAPH 2003 Papers on - SIGGRAPH '03
We describe a generalization of the method of barycentric coordinates for planar parameterization which solves the spherical parameterization problem, prove its correctness by establishing a connection  ...  Parameterizing a triangle mesh onto the sphere means assigning a 3D position on the unit sphere to each of the mesh vertices, such that the spherical triangles induced by the mesh connectivity are not  ...  Generalizing the barycentric coordinates theory to spherical embeddings is not straightforward.  ...

Generalized barycentric coordinates and applications

Michael S. Floater
2015 Acta Numerica
This paper surveys the construction, properties, and applications of generalized barycentric coordinates on polygons and polyhedra.  ...  Spherical coordinates Spherical barycentric coordinates for points in a spherical triangle were studied by Möbius (1846, Sec. 14) and Alfeld, Neamtu and Schumaker (1996) , and used in the latter to  ...  Unlike in the case of a planar triangle, spherical barycentric coordinates do not in general sum to 1.  ...

Research on Goods and the Ship Interaction Based on ADAMS

Fangzhen Song, Yanshi He, Haining Liu, H.L. Yuan, R.K. Agarwal, P. Tandon, E.X. Wang
2017 MATEC Web of Conferences
Figure 2 . 2 Barycentric coordinates curve. Figure 3 . 3 Barycentric coordinates curve.The rigid body under the constraint of barycentre trajectory is shown inFig. 4.  ...  Get the change of barycentric coordinates as shown in Table 1: Table 1.  ...

The Celestial Reference System in Relativistic Framework

Han Chun-Hao, Huang Tian-Yi, Xu Bang-Xin
1990 Symposium - International astronomical union
The concept of reference system, reference frame, coordinate system and celestial sphere in a relativistic framework are given.  ...  The problems on the choice of celestial coordinate systems and the definition of the light deflection are discussed. Our suggestions are listed in Sec. 5.  ...  It is neither stationary nor spherically symmetric.  ...
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